OK, I did the tutorial, got my 480 or whatever dilithium. Now I'm doing it again for real and here's what's going on.
I enter the Kahless expanse, find the first Fed fleet, defeat the 6 battle groups, then leave. I need to find two more, so I scan signals until I find the next one.
This time there's no counter for how many group kills I need. I've been here killing ships for maybe an hour, if not longer, and the quest still isn't updating. I've HAD to have killed 6 groups by now.
Even without this apparent glitch, this seems like a hell of a lot of trouble compared to the Fed exploration missions. Should be 3 groups for every fleet you find.
You don't have to leave the encounter after you finish the first set, just wait for it to reset. It works like the non-Borg Deep Space Encounters. Don't bother killing anything until the counter is up for the next set.